Decomposing finitely generated groups into free products with amalgamation

The problem of the existence of a decomposition of a finitely generated group Gamma into a non-trivial free product with amalgamation is studied. It is proved that if dim X-s(Gamma) greater than or equal to 2, where X-s(r) is the character variety of irreducible representations of Gamma into SL2(C), then Gamma is a non-trivial free product with amalgamation. Next, the case when Gamma = (a,b \ a(n). = b(k) = R-m(a,b)) is a generalized triangle group is considered. It is proved that if one of the generators of Gamma has infinite order, then Gamma is a non-trivial free product with amalgamation. In the general case sufficient conditions ensuring that Gamma is a non-trivial free product with amalgamation are found.